Optimum system of symmetry algebra and classification of exact solutions of the flow energy equation.

Author
Mehdi Jafari
university
Abstract
The Lie symmetry method, first developed by Sophus Lie, is a fundamental tool in the analysis of differential equations, particularly for generating exact solutions and reducing equation order. One of its key applications lies in identifying invariant solutions and simplifying complex nonlinear systems through symmetry reductions. In recent decades, this method has been extended and applied to a wide range of physical models. The present study focuses on the flow energy equation, which arises in fluid mechanics and describes thermal energy distribution in incompressible Newtonian pipe flow. Using classical Lie point symmetries, we obtain the symmetry generators of the equation and classify its one-parameter subalgebras through the adjoint representation. Based on this classification, an optimal system is constructed to systematically generate non-equivalent invariant solutions. In addition to symmetry reductions, we compute several exact solutions of the reduced equations and analyze the structure of conservation laws associated with the flow energy equation. These conservation laws are derived using both direct and variational approaches. The results highlight the effectiveness of Lie symmetry analysis in understanding and solving nonlinear PDEs in applied mathematics and physics.
Keywords
Symmetry Lie group of equations
flow energy equation
conservation laws
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